The Edinburgh outfit had been seeking the double, but now they will have to be content with aiming for the Premiership crown.
New Zealand Test star Papps had laid down the foundations with a superb century - but it was when he bowled the final over of an absorbing contest at Grange Loan that he really excelled.
The hosts - needing just four runs for victory - just couldn't do the business when it mattered most and went down by a single run.
Former skipper Stevie Gilmour grabbed two from the first ball of the over, then played out a dot-ball from Papps.
The pendulum swung when he was dismissed with the next delivery - passing the responsibility to Ali Evans.
Evans had to block out the next two balls and the Ayr fielders reacted in jubilant style as he was run out from the next delivery.
The importance of Papps was underlined by the fact that he was the only Ayr batsman to pass the 30-mark in their tally of 188.
Evans was the pick of the Carlton attack, snapping up four scalps to warm up nicely in the build-up to the Scots' Intercontinental Cup tussle with the Netherlands at Mannofield this week.
Ayr were buoyed up when they removed star turns Cedric English and Fraser Watts in the early stages of the reply, Scotty McElnea clean bowling English then trapping Watts lbw.
It could have been even better for Ayr had Alan Davidson not spilled a chance offered by Preston Mommsen.
Mommsen (43) and Gilmour (53) appeared to have done enough to steer Carlton past the finishing post. But then came the all-important over from Papps, although McElnea played a massive part in the upset with four for 30 from his ten overs.
Holders Uddingston had old stagers Paul Hoffmann and Ricky Bawa to thank for their comprehensive victory over Falkland at Scroggie Park. Hoffmann (66) and Calum MacLeod (67) propelled the Lanarkshire brigade to 248.
And when the weather took control of matters, the Fifers had responded with 72 for seven - 122 adrift of the rate.
Captain Bawa led the assault, bagging four for 26.